Page 53 - Modern physical chemistry

42 Gases and Collective Properties

Quantity k is called Boltzmann s constant. The gas constant R equals Avogadro's number
times k, by definition; while N equals Avogadro's number times the number of moles n.
So (3.22) may be rewritten in the form

E tr =~RT. [3.23]
n 2
Eliminating E tr from (3.15) and (3.23) leads to the ideal gas law

PV=nRT. [3.24]
A common laboratory thermometer consists of a glass bulb containing mercury
attached to a uniform capillary tube. In operation, the surface of the mercury moves up
or down the tube as the mercury expands or contracts more than the glass. The distance
the column moves on heating the bulb from the freezing point of water to the boiling
point at 1 atm pressure is determined. One hundredth this distance is taken as the celsius
unit C and a linear scale is constructed. The freezing point of water is labeled 0° C; the
boiling point at 1 atm, 100° C.
By experiment, it is found that the volume of a gas varies linearly with the mercury-
in-glass celsius temperature t to a good approximation,

[325]

when the pressure P of the gas is kept fixed at a low value. But equation (3.24)
rearranges to
V= nR T . [3.26]
P
Thus, the ideal-gas temperature T varies linearly with the t in (3.25). If we take the unit
of temperature the same, we have
T=t+To· [3.27]
The unit for this T is called the kelvin K.

3.4 Constants R and To

The gas constant R and the ice point To can be determined from limiting measure-
ments on gases.
The pressure-volume product for a real gas deviates from (3.24) because of interac-
tions among its molecules. But whenever its pressure is lowered, the average distance
between the molecules increases and this interaction decreases. As the pressure goes to
zero, the behavior of the gas approaches that of an ideal gas. At a given temperature,

lim PV -(PVJ [3.28]
p--.o n - n ideal •
Combining this result with (3.24) yields

lim PV =RT. [3.29]
P--.O n
Here P is the observed pressure, V the measured volume, n the number of moles of the
gas, and T the kelvin temperature.